# Vertex Definition (Linear optimization)

I don't understand a definition of the vertex over a convex polyhedron in standard form :

$P=\{x\in\mathbb{R}^n, Ax=b, x \geq 0\}$

$x$ is a vertex of P if and only if the columns $\{A^j\in \mathbb{R}^m:x_j > 0\}$ of $A$ are linearly independent.

What is this set : $\{A^j\in \mathbb{R}^m:x_j > 0\}$ ?

$\{A^j\in \mathbb{R}^m:x_j > 0\}$ is the set of columns of matrix A for which corresponding components of vector $x$ are positive.

• Thanks, not sure if I understood it at the time and not sure how you got to dig this question out. :D Have a nice weekend. – matovitch Oct 13 '17 at 13:37